A computer investigation has been made for the problem of three coherent waves in a plasma system assuming a nonlinearly unstable situation and mutually different linear damping of the three interacting waves. The results agree satisfactorily with those of an analytic approach to the problem (Wilhelmsson 1970). In the present paper the solution near the explosion time has been further examined.

A solution of the one-dimensional time-independent Vlasov–Poisson system, including the condition of no net current, is constructed for a large class of electric potential functions φ(x), including periodic and solitary waves, as well as monotonic transitions. Displaced Maxwell distributions for the free particles, and an unshifted Maxwell distribution for the trapped ions, are used. The condition of positiveness of the unknown distribution function for the trapped electrons (which can be split into two parts, one depending solely on the chosen wave, the other on the given distributions) is examined. It is shown, for a non-linear wave

that this condition plays no rôle when the thickness l of the wave considerably exceeds the Debye length λD(l ≫ λD). If, however, l lies in the neighbourhood of λD(l ≳ λD), there exist limits for the free parameters. The smallest thickness lmin results if the Mach number M lies in the range 1 > M > 2 and the electronion—temperature ratio θ = Te/Ti exceeds 10. This confirms the view that the wave is a steepened ion-acoustic wave. lmin decreases with decreasing amplitude ϕmax of the wave and decreasing number of trapped ions, but does not lie below the Debye length as long as the wave is non-linear. In the linear case, the condition of positiveness imposes no restrictions.

The instability of the ordinary electromagnetic mode propagating perpendicular to an external magnetic field is studied for two counter-streaming plasmas with anisotropic temperatures. A necessary and sufficient condition for this instability is derived and the growth rate is calculated. Although instability can occur in a plasma with low β‖e, provided the streaming velocity is comparable to the parallel electron thermal velocity, β‖e of the order of unity and large electron temperature anisotropy (T‖e≫ T⊥e) are required for large growth rates. These large growth rates are typically of the order of the electron cyclotron frequency. Comparison with the electrostatic two-stream instability is made.

The distortion of the electron velocity distribution caused by a large amplitude Landau wave is determined analytically for the initial-value problem. The resulting stability of electrostatic perturbations impressed on the evolving plasma is studied. Narrow sidebands of the applied frequency experience consecutive growths of large magnitude during the early stages of the nonlinear wave-particle interaction. The significance of the derived results to both wave propagation experiments and triggered VLF emissions in the magnetosphere is discussed.

A treatment is given of the problem of constructing normal modes for anarbitrarily bounded system from roots of the linear dispersion relation D( ω, k) = 0 for the corresponding infinite or periodically bounded system. For a system described by continuous macroscopic variables, and of general cylindrical form (uniform along an axis z, say), each transverse eigenmode gives rise to a set of axial normal modes constructed from a pair of dominant roots kαz(ω) of D = 0 satisfying the boundary conditions which are characterized by complex reflexion coefficients for the dominant waves. The implications of the results for the interpretation of experiments on plasma waves and instabilities on finite cylinders are discussed, with particular reference to the effects of end-plate damping and axial current on Q-machines.

Using the water-bag model of an electron plasma the nonlinear interaction of two plasma wave pulses (each of sufficiently narrow wave-number spectrum) is discussed. It is shown that, for the special case of a resonant (zero group velocity) pulse interacting with a non-resonant pulse, the interaction produces an effective slowing down of the group velocity of the non-resonant pulse. An application of this theory to the explanation of the apparent resonances at fractions of the plasma frequency observed in the ionosphere is discussed.

In this paper we consider the interaction between waves and particles in a weakly turbulent, collision-free plasma for the case of pure Coulomb interaction and zero magnetic field. Starting with the Vlasov equation we suggest a solution by means of a perturbation expansion and a diagram technique, which classifies the various interaction processes and leads to the derivation of kinetic equations.

The work of Butler & Gribben (1968) on a general formulation ofthe problem of non-linear waves in non-uniform plasmas is extended. The particular case treated in Butler & Gribben, § 6, is discussed again (the distribution functions fs and electrostatic potential depend only on one space co-ordinate, that in the direction of propagation of the wave, the wave is slowly varying only with respect to this co-ordinate and time, the magnetic field vanishes and relativistic effects are negligible). Conditions necessary to avoid secular terms in the solutions of the Vlasov and Poisson equations are rederived, but without resorting to assumed series expansions in e, the perturbation parameter describing the non-uniformity, for the dependent variables, and using a different notation which simplifies and retains the symmetry of the equations. The corresponding general boundary condition, to be satisfied by fs innergy space at the boundary of the trapped particle region, is also derived.

Particular attention is devoted to the Vlasov equation, and the derived general result is used to obtain the appropriate necessary condition from this equation, correct to O(ε). This shows that, unlike the leading-order theory, at the next stage an extra length scale appears in the equations, which would be needed if e.g. the theory were to be used to discuss shock waves. It is argued that this result, taken with the mixing process described in Butler & Gribben (which supplies a mechanism for the formation of shocks), appears to place the theory at least on as satisfactory a level as the Navier-Stokes theory for the discussion of shock wave structures.

Another aspect of the analysis is the comparison with the Whitham (1965a) averaging method for treating the propagation of non-linear waves in other fields. Similar features are pointed out, and it seems likely that the averaging method is equivalent to the leading-order theory. It is thought that the present approach might prove to be useful in other wave problems.

Nanosecond pulses are launched, and propagate on a positive column in the form of electron waves. The dispersion of the pulses is studied experimentally, and is found to be in agreement with monochromatic experiments and theory. Apart from wave attenuation, which is observed to depress signals of higher frequencies, Fourier analysis of the received pulses has revealed that the launching mechanism favours frequencies in the vicinity of 1/6ωp A full-wave theory, based on an extension of Goubau's excitation theorem, is derived, and successfully contrasted with experiment.

A new method for dealing with a typical quasi-linear problem is presented. The quasi-linear relaxation of the temperature anisotropy of a collisionless plasma is completely described within the framework of the thermodynamics of irreversible processes. This is obtained by means of generalized Onsager relations applied to the weakly unstable system. Free energy of the system is analyzed in detail, to study the net work involved in the process.

This paper considers electrostatic waves in a Vlasov plasma of unmagnetized ions and magnetized electrons undergoing E x B and gradient B drifts. The linear dispersion relation is solved numerically for Te & Ti. The results show that, as in the Te <Ti case, increasing electron β decreases the maximum growth rates.

A model kinetic equation for the internal fluid of diatomic molecules which interacts with thermal radiation is proposed. The cross-collision term developed for the molecule-photon interaction has the property that molecules and the sum of internal and photon energies are conserved. An alternative approach to this term based on the product of two BGKW collision operators yields the same result. It is also shown that the proposed model leads to an H-theorem.

The mathematical treatment of the influence of a weak coupling on an assembly of damped oscillators with different eigenfrequencies permits us to exhibit a peculiar property of the solution: namely, the existence of a high frequency solitary, weakly damped mode. We propose to apply these results to the ionic oscillations of a bounded plasma.

The influence of the cathodes of a double-ended Q-machine on a flute instability of the Rayleigh–Taylor type is investigated for a low β plasma column. It is shown that flute modes are strongly affected by the end plate conditions and by the machine length. The occurrence of ion-rich sheaths is a necessary condition for the validity of the dispersion relation of the unbounded case. The range of values of the column length for the existence of a given flute mode is determined.

The neoclassical diffusion in a tokamak is calculated for a finite-β plasma. The flux of the particles is shown to be substantially unchanged, with respect to the low-β case. The ‘equilibrium’ equation is derived. From it the magnetic surfaces can be obtained, once the external electromagnetic field and the density and temperature profiles are given. The case of magnetic surfaces with circular cross-section, in the small aspect ratio limit, is considered in detail.

We study the effects of circularly polarized plasma waves in anisotropic homogeneous plasmas. The waves propagate in the direction of an external uniform magnetic field. It is shown that in many cases only one mode, with wavenumber k, is present in the plasma. We present a computer model suitable for the study of such cases. The model is used to simulate linearly stable and unstable phenomena. A good agreement with previous predictions of linear theory is obtained. Nonlinear effects are studied and mechanisms that stop instabilities are described.

A study of the compressive magneto-acoustic waves in a guiding centre plasma shows that the wave-front that emerges from a point disturbance after a finite time is a simple oblate spheroid with the axis of revolution parallel to the field lines. Thus, in a steady three-dimensional supersonic flow of guiding centre plasma a simple analytic expression can be obtained to represent the characteristic surfaces. From a proper linear combination of the governing macroscopic equations, the characteristic equation is obtained. It represents the propagation of disturbances on the characteristic surface. The characteristic theory can be used to study the interaction of the solar wind with the moon and possibly with other planetary bodies.

We present an experimental investigation of electron cyclotron resonance plasma accelerators as a source of high-intensity, low-energy (keV) ions in which ion current and electron and ion energies are related to r.f. power and gas pressure.

A Lagrangian, for the slowly varying complex amplitude of an almost monochromatic electrostatic plasma wave in an unmagnetized plasma, is derived. The method is a variant of the averaged Lagrangian technique of Whitham, adapted for use in plasma physics by employing Low's Lagrangian, together with certain elimination procedures to simplify the Lagrangian to usable form. The expansion in powers of the wave amplitude is carried out to quartic terms, and non-adiabatic terms are also retained. Variations with respect to the amplitude lead to a nonlinear Schrödinger equation for the amplitude, while variations of the particle trajectories lead to a modified Vlasov equation, which includes the nonlinear reaction of the wave on the average trajectories. It is assumed that there are no resonant particles. These equations are coupled through the nonlinear frequency shift. It is found that the particle aspect of the plasma has a profound effect on the stability of the system, due to resonance of the wave envelope with particles moving at the group velocity. This effect, nonlinear Landau damping, is always destabilizing, leading to growth of modulations. In terms of the nonlinear Schrodinger equation, the effect changes the equation from a Hartree-Fock equation with a delta-function interaction to one with a non-momentum-conserving, non-local interaction. In the limit of fairly small wavelength, of the modulations, it is shown that the growth rate approaches that expected from plasma kinetic theory.

It is shown that the effect of the cut-off criteria of the electronic partition functions on transport properties varies with the particular coefficient considered. For example, the cut-off has little effect on translational thermal conductivity, but a large effect on viscosity and reactive thermal conductivity.